Find the solution of the following equation whose argument is strictly between $270^\circ$ and $360^\circ$. Round your answer to the nearest thousandth. $z^4=-625$ $z$ =
Answer: The Strategy A straightforward way to solve an equation of the form $z^{n}=m$ is by using the polar form of $z$. Therefore, our solution will consist of the following steps: Rewrite $z^n$ and $m$ in polar form. [How is this done, in general?] Solve for the modulus and argument of $z$. Find the rectangular form of $z$. [How is this done, in general?] Rewrite the equation in polar form Let's denote $r$ and $\theta$ to be the modulus and argument of $z$, respectively. Therefore, $z^{4}=r^{4}[\cos {({4}\cdot\theta)}+i\sin {({4}\cdot\theta)}]$. The number $-625$ has a modulus of $625$. The argument of $-625$ can be $180^\circ$ plus any multiple of $360^\circ$, so we can write it as $180^\circ+k\cdot360^\circ$ for an integer $k$. Now the equation looks as follows: $\begin{aligned}r^{4}[\cos {({4}\cdot\theta)}+i\sin {({4}\cdot\theta)}]&= \\625&[\cos(180^\circ+k\cdot360^\circ)+i\sin(180^\circ+k\cdot360^\circ)]]\end{aligned}$ When two complex numbers are equal, we know that both their moduli and arguments are equal. Therefore, we have the following equations for $r$ and $\theta$ : $r^{4}=625$ ${4}\cdot\theta=180^\circ+k\cdot360^\circ$ Solving for $r$ $\begin{aligned}r^{4}&=625 \\\\ r &=5 \end{aligned}$ Solving for $\theta$ $\begin{aligned}{4}\cdot\theta&=180^\circ+k\cdot360^\circ \\\\\theta&=45^\circ+k\cdot90^\circ\end{aligned}$ Remember that $\theta$ is strictly between $270^\circ$ and $360^\circ$. Therefore, we need to find the multiple of $90^\circ$ that is strictly within the range of $270^\circ-45^\circ=225^\circ$ and $360^\circ-45^\circ=315^\circ$. This multiple is simply $270^\circ$, so $\theta=315^\circ$. Finding the rectangular form of $z$ Let's plug in $r=5$ and $\theta=315^\circ$ into the polar form of $z$ : $\begin{aligned}z&=r[\cos(\theta)+i\cdot\sin(\theta)]\\\\ &=5[\cos(315^\circ)+i\cdot\sin(315^\circ)]\\\\ &=5\cos(315^\circ)+5\sin(315^\circ)\cdot i\end{aligned}$ Using the calculator and rounding to the nearest thousandth, we get the following solution: $z=3.536-3.536i$ Summary $z=3.536-3.536i$